Skip to main content

The behaviour of random discrete structures at criticality

Final Activity Report Summary - CRITICAL STRUCTURES (The behavior of random discrete structures at criticality)

How long does it take the members of a reproducing and migrating population to send its most far-flung members to the other side of the world? How far into the bedrock does groundwater seep? How many links can it take to get from one web page to another? These three seemingly distinct questions share in fact a profound similarity, since they are all about extreme distances in settings which contain randomness or unpredictability, or else ‘random structures’.

Our project focussed on extreme distances, particularly on random discrete structures. Though there are many possible ways to formalise the sense in which those questions are similar, it turns out that many of these formalisations share common features. In particular, it is extremely common to see the appearance of a ‘phase transition’, which is a point at which the behaviour of the system changes drastically, e.g. from typical chains of links between websites being extremely long or nonexistent to the sudden existence of short paths between most websites. We were particularly interested in the moments at which these changes, the so-called ‘critical phenomena’, occurred, specifically as they related to distances.

We made substantial progress on understanding such critical phenomena. In particular, we succeeded in establishing a ‘metric space’ limit for the critical Erdos-Renyi random graph. The Erdos-Renyi random graph was one of the best-studied discrete models of random structures, nevertheless information about distances in Erdos-Renyi random graphs proved surprisingly tricky to come by. Our limit result answered a whole class of questions about typical and extreme distances in such graphs. We also answered relevant questions about the typical behaviour of smooth functions on such random graphs. These results could be seen, for example, as providing evidence on how different languages in extremely distant countries could be if pairs of neighbouring countries always had similar, but not necessarily identical, languages.